WASHINGTON — The top two Republicans in Congress said on Monday that they supported investigations into possible Russian cyberattacks to influence the American election, setting up a potential confrontation with President-elect Donald J. Trump in his first days in office.

“Any foreign breach of our cybersecurity measures is disturbing, and I strongly condemn any such efforts,” said Senator Mitch McConnell, Republican of Kentucky and the majority leader, adding, “The Russians are not our friends.”

Mr. McConnell’s support for investigating American intelligence findings that Moscow intervened in the election on Mr. Trump’s behalf could presage friction between the Republicans who control Congress, and who have long taken a hard line against Russia, and the president-elect, who has mocked the findings.

Mr. McConnell also went out of his way to address Mr. Trump’s claim that the C.I.A. could not be trusted because of flawed intelligence before the Iraq war.

“Let me say that I have the highest confidence in the intelligence community,” Mr. McConnell said, “and especially the Central Intelligence Agency. The C.I.A. is filled with selfless patriots, many of whom anonymously risk their lives for the American people.”

The top Republican in the House, Speaker Paul D. Ryan of Wisconsin, said he supported a continuing investigation by Representative Devin Nunes of California, the chairman of the House Intelligence Committee. In a statement, Mr. Ryan said: “As I’ve said before, any foreign intervention in our elections is entirely unacceptable. And any intervention by Russia is especially problematic because, under President Putin, Russia has been an aggressor that consistently undermines American interests.”

Congressional Republicans announced their support for inquiries after Mr. Trump railed for much of the weekend against the intelligence findings. But their remarks, especially Mr. Ryan’s, were far from fiery, reflecting both a fear of offending Mr. Trump, who has taken many positions against traditional Republican orthodoxy, and the Republicans’ belief that Democrats have selectively leaked intelligence information for political gain.

Critics from both parties are questioning Mr. Trump’s apparent choice of Rex W. Tillerson, the chief executive of Exxon Mobil, as secretary of state, particularly because of his longstanding business connections with Russia and his close relationship with President Vladimir V. Putin, whom he has known for two decades. Mr. Trump said in a Twitter post on Monday night that he would make a formal announcement on the job on Tuesday morning.

Senators Lindsey Graham of South Carolina and Marco Rubio of Florida, both Republicans, have expressed concern about the reports of cyberattacks, as have numerous Democrats. But Mr. Rubio, in an apparent reference to Mr. Tillerson, went a step further on Monday, writing on Twitter, “Being a ‘friend of Vladimir’ is not an attribute I am hoping for from a #SecretaryOfState.”

Mr. McConnell said the Senate investigation would be led by Senator Richard M. Burr, Republican of North Carolina, the chairman of the Intelligence Committee. Senator John McCain, Republican of Arizona, the chairman of the Armed Services Committee, will add a subcommittee to look into cyberattacks, led by Mr. Graham.

“The first thing we want to establish is, ‘Did the Russians hack into our political system?’” Mr. Graham said in an interview on Monday. “Then you work outward from there. I have a high degree of confidence Russia did this.”

Mr. Nunes, a member of Mr. Trump’s transition team, said in a statement that the Intelligence Committee had been “conducting vigorous oversight of the investigations into election-related cyberattacks.”

Democrats have used the latest intelligence findings to renew their calls for an urgent inquiry. John D. Podesta, Hillary Clinton’s campaign chairman, demanded on Monday that all information about Russia’s meddling be declassified, and that the Obama administration explain what it knows about the hacking and when it knew it.

“We now know that the C.I.A. has determined Russia’s interference in our elections was for the purpose of electing Donald Trump,” Mr. Podesta wrote in a statement. “This should distress every American. Never before in the history of our republic have we seen such an effort to undermine the bedrock of our democracy.”

Three Senate Democrats — Benjamin L. Cardin of Maryland, Dianne Feinstein of California and Patrick J. Leahy of Vermont — called on Monday for the creation of an independent, nonpartisan commission to comprehensively investigate allegations of Russian interference in the 2016 election.

But Mr. McConnell stopped short of calling for a special select committee, saying that the Senate Intelligence Committee was “more than capable of conducting a complete review” of the matter.

While he stopped short of saying whether he agreed that Russia had interfered in the election in support of Mr. Trump, Mr. McConnell said, “We need to approach all these on the assumption the Russians do not wish us well.”

Mr. McCain was less equivocal, saying Monday that there was “no doubt about the hacking” by Russian intelligence services. He called the hacking of the Democratic National Committee and related accounts “another form of warfare” in an appearance on “CBS This Morning” with Senator Chuck Schumer of New York, the incoming Democratic leader.

For his part, Mr. Trump was dismissive of the intelligence findings and suggested that Democrats were simply stirring controversy. “Can you imagine if the election results were the opposite and WE tried to play the Russia/CIA card. It would be called conspiracy theory!” Mr. Trump said in a Twitter post on Monday.

The White House press secretary, Josh Earnest, said that the administration would support a congressional review. He also rejected the notion that the administration had failed to adequately highlight the Russian efforts before the election, saying it had extensively briefed Congress all year about Russian electoral meddling.

“There has been intensive cooperation between the intelligence community and other national security agencies, and members of Congress in both parties, both before and after the election,” Mr. Earnest said. “The briefings have been provided in a variety of settings, both classified and unclassified.”

Even beyond the conclusions of the intelligence community, Mr. Trump’s campaign had widely known and extensive ties to the Russian government. A campaign manager, Paul Manafort, had worked for the Russian-backed government in Ukraine, and Mr. Trump’s choice for national security adviser, Lt. Gen. Michael T. Flynn, had consulted for a Russian-backed media group, Mr. Earnest noted.

Mr. Earnest said that Congress had a “special responsibility” to investigate the ties between the Trump campaign and the Russian government, because those connections were widely known before the election. He added that, for Capitol Hill Republicans, how to “reconcile their political strategy and their patriotism is something they’re going to have to explain.”

Gardiner Harris and Jonathan Martin contributed reporting.

A version of this article appears in print on December 13, 2016, on Page A1 of the New York edition with the headline: G.O.P. Feud Looms as Leaders Back Russia Inquiries. Order Reprints|Today's Paper|Subscribe